It would have solved all problems had the photo been shot from the RH side as I believe the badge for both Triumph and BSA was on the RH side of the tank.
Both BSA and Triumph had badges on both sides of the tank.
However in the case of Triumph the badge after circa 1950 had chrome embellishments.
Triumphs also had separate speedos or separate matched speedo & rev counters.
The first thing I noticed from the picture was the bike had been dropped at least once hence the bent clutch lever.
Cheers
Guy